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The position of boom $A B C$ is controlled by the hydraulic cylinder $B D .$ For the loading shown, determine the force exerted by the hydraulic cylinder on pin $B$ when $u=65^{\circ}$.

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Video by Donald Albin

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Physics 101 Mechanics

Chapter 10

Method of Virtual Work

Work

Potential Energy

University of Washington

Hope College

University of Winnipeg

Lectures

02:08

In physics, work is the transfer of energy by a force acting through a distance. The "work" of a force F on an object that it pushes is defined as the product of the force and the distance through which it moves the object. For example, if a force of 10 newtons (N) acts through a distance of 2 meters (m), then doing 10 joules (J) of work on that object requires exerting a force of 10 N for 2 m. Work is a scalar quantity, meaning that it can be described by a single number-for example, if a force of 3 newtons acts through a distance of 2 meters, then the work done is 6 joules. Work is due to a force acting on a point that is stationary-that is, a point where the force is applied does not move. By Newton's third law, the force of the reaction is equal and opposite to the force of the action, so the point where the force is applied does work on the person applying the force. In the example above, the force of the person pushing the block is 3 N. The force of the block on the person is also 3 N. The difference between the two forces is the work done on the block by the person, which can be calculated as the force of the block times the distance through which it moves, or 3 N × 2 m = 6 J.

03:23

In physics, mechanical energy is the sum of the kinetic and potential energies of a system.

31:04

The position of boom $A B …

06:24

The position of member $A …

03:30

06:56

I'm gonna draw a diagram of what's going on here. We've got a ride. There are eight Kipps. That's killer pounds. 8000 pounds. Here we have a force at B, so I'm gonna put a be See. This force is in this direction where you have a point d up here, which is a distance of to, um So I'm gonna call this de and I'm going to say that it equals two feet. All right. Um, Fada is here. I'm gonna call this L We're l is 1.5 feet, and so I'm gonna call this to our all right? I'm going to set up my coordinate system so that X and Y are here, So why not? A is positive three l and ah, sign is opposite overhype oddness. That would be signed, fada Um, the y value at B actually gonna need the y value and the X value would be the Y. Value at B is going to be too well sign Fada and the X value at B is, um to l co sine theta so I can take the derivatives delta. Why a is three l cause I Anthea Delta, why be is too well sine theta and X Actually, actually, negative was negative to l co sign data derivative of the coast on is the opposite of the sign. So Delta X b I was going to be to l sign data, and I forgot to rate my delta status. Delta Fada, Delta fade. All right, now I need to figure out the force, which I'm going to call. I'm gonna call this P here, so I'm gonna call this Q. So cue in the X direction is going to be que times something. So let's try to figure out that something if I make a horizontal line here and if that's too, um All right, let's So this is 90 and this would be 90 minus fade. Hm. I'm thinking about the law of co signs. Uh, and if I had the law of co signs, I could get this length here. And if I had that length, I could use the law of signs to get this angle. I'm just trying to decide if there is easier way. Um, if I use the Pythagorean, the're, um I could figure out this length, and then I could figure out the upper length. I couldn't figure out that length a man. So the Pythagorean Theorem methods gonna be leaving a little longer. All right, so let's just use the law of co signs. Uhm, I'm going to call this distance up here. Um, de too so d a two squared equals su el Squared plus d squared to l squared plus d squared minus to Hell de co sign Fada. And I'm gonna do this Going to write that? Okay, so I know d to it's too. Now, let me just double clutch check here as we look at the law of co signs. Um okay, so I already had a two here is actually two twos. It's ah, see school because a squared plus B squared minus two a b co sign. See? So I need another two. A be co sign c. All right, so now, um, the sign of that green angle. So I'm gonna call fi over. De is going to eke wall the, um, sign of fada over de sub to. And so now I can get five And FYI minus 90 minus data, which would be the same uh, fyi minus 0 90 minus stated There we go. That's what I was thinking. So that would be the same as by plus feed. Ah, minus 90. That is this little angle in here, and I'm running out of colors. I'm gonna use red. FYI, I can't read it. That small call that gamma Gamma is five plus state of minus 90 by plus data minus 90. Okay, so cue in the X direction is just Q co sign GAM and cue in the Y direction is just Q sign gam, and they are both positive. It's going positive in the extraction and positive in the Y direction. All right, so, um, virtual momentum equation changing potential energy zero, which is p which is negative times. Delta y dolar y is positive three l co signed a $2 theater. Okay, now we've got let's do que why? Which is que signed gamma times delta y to elves sine theta delta theta. Um And then we've got, um, Hugh X, which is Cuco sine gamma times Delta X, which is to l sine theta Delta Zeta. All right, the doll. If Data's cancel out, do the els cancel out. Yes, they dio l No. Yeah. Okay, So now I have PICO sine theta. Now there's a three there also three p Hussein Fada equals que times two. There's a two in both of those two. Q sine gamma plus co sign GAM Signed Gamma Plus CO sign Gamma. Now I'm just making sure that I didn't make any mistakes and I definitely did, because I'm missing some fate as here. So let's it's erase that. Definitely missing something here. It's tu que sine gamma sign theater plus tu que co sign gamma signed data. But if I'm going to write that, I might as well factor out the sign data. But why do I have a scientist? A. Did I put the wrong thing in? Um, okay, Q X is Cuco signs. This is the X one over here. So Delta X is going to be Ah, this This is where I missed it. Derivative of the sign is the co sign right here noticing that there was a problem. Okay, so the y value, which would be this one, needs to have a co sign. And so now it's signed Gamma, the race co sign data and then co sign gamma signed data. Okay, this looks better, but I did want to double check to make sure that I have my negatives and positives correctly. Correct. So the y value point A is positive. That's good. The Y value point B is also positive, and the X value at point B is neg Good. When you take the derivatives of that, you get this all right. P is negative. And I believe that I indicated p to be negative. Yes, Q X and Q. Why are both positive? And I indicated there was positive. Now utilizing a trigger no metric identity sign Alfa Co sign Data plus co sign Alfa sine theta is just, um, the sign. Ah, that's not Alfa. That's gamma plus data. So three p co Zain fada equals two Que sine gamma plus data. But gamma is five plus beta minus 90 Sign Ah, gamma is five plus state of minus 90. By plus, they'd ah minus 90 plus fada. Okay, so that is the sign of FYI plus tooth Ada minus nine. Just write everything out again. Okay? But no, we have up here that the sign of Phi is de over d to signed Fada. Let's move that down here de over D to sign. Fada. FYI is de over D to science data, but D two is the square root of two l squared plus the square root of D squared. All right, try to conserve the room. Here to l squared is just four l squared plus d squared minus for L D co sign Fada. All right, I'm just gonna move this. Okay? So if I So for this substituted in there and then graph it in Dismas dot com and then I look for the X intercepts. That should be the answer. So three he he is. Oh, P is what we're solving for, So I'm going to write. Oh, and you know what? If I'm solving for P, then let's just do another step here. P is to que over three, cause I ninth ada sign of Phi plus Tooth Ada minus nine. Okay. And so now this is a little bit simpler. I can just do two times. Q Okay. What is que Oh darn it. We're solving for Q not not p a man. Okay, well, let's try this again. Solving for Q not p que is three p co signed Theda over to sign five plus tooth Ada minus 90 degrees. Okay, now let's try it. Que is 8000 pounds. No. P is £8000 3 times 8000. I'm the co sign Now we're told to use a fada of 70 degrees. I'm gonna set my calculator for degrees. No sign of 70 over to Fine. Oh, to say tha That's two times 70. Minus 90 plus I. But FYI is the inverse sign. Um uh, De got some D's and l's here. D is to l is 1.5 these two over square root for 1.5 squared plus two squared minus four times 1.5 times two co sign of 70. Okay, so now I've got may Think about what I've been doing here. Three p co sign 70 over to sign of. I did, too. Fate of first minus 90 plus Phi and Phi is the inverse sign I will all that stuff to. Okay. And then I have to put my sign data in there signed 70. It's strange. It seemed like it was the same answer. Whether had that sign? 70 or not, it's very close. We'll check them into in degrees. Okay, It's giving me an answer of 4104. 3 4105 pounds. And I'll be impressed, if that is correct. Ah, there's no answer in the back of the book for such a difficult question. There's no answer in the back of the book. Oh, man, 4105 pounds is my answer. All right, So first, does my answer make sense £4105 at B is going to make up for £8000 a point? P. Um, that doesn't make sense to me. I would have thought that the force would be greater at Point B. However, fellow the downward force is relieved by point see, because point C is constrained vertically. So now I'm thinking that this 4000 105 does make sense. So then, after I decide whether my answer makes sense or not, and by the way, it's the rate this it's Lisa's positive, and I thought it should be positive. Um, so it makes sense. So the second question is, did I make any calculation mistakes? Now? As I was working, I was double checking my work um but I even caught some mistakes as I was going through, so there may be more mistakes. So what I would dio is I would go through every single thing. This Why is three El co sign fade? No sign that that makes sense. Ah, derivative would be Rielle derivative of Sinus coastline data. That makes sense. I want to look at my diagram to Did I write anything wrong? De is, too, If I call that Alan too. Well, then, l would be 1.5. Got data in the right place. 90 minus data would be there. Okay, Tooele sine theta and negative to Elko. Sign data. That seems pretty simple. So I did. The derivatives driven coast on is the opposite sign. Did that correctly? This green stuff here seemed like the most difficulty I used. The law of co signs d two squared equals to l squared plus d squared minus two. I almost forgot the to times to l Times d times a co sign data that all makes sense. And then give me d to then the sign of Fada over De too. Because the sign of Phi Over De So that makes sense. But then looking here where gamma is, If I took 90 minus, they'd Ah, no. If I took fine, then I subtracted off 90 minus data that would give us camp Take fi, distract off 90 minus data. So five plus data minus 90 gives gamma. So that looks good. Que In the X direction is Cuco sign data? Yep. Why direction would be signed? Data science data Their sign Phi is de over d to sign. Fate of that makes sense. I did double check already that I had my positives and negatives Correct Here. So three p. Hussein thinking I equals two Q sine Gamma Co sign Data Hussein Gamma Sign Theater to sign Gamma plus data, which is the sign. Gamma is five plus state of minus 95 plus state of minus 90 plus data three. PICO Cynthia five plus two data minus 90 And then this was from above. Sign of Phi is de over D to sign data de over D to sign data but D two is the square root of four l squared plus d squared. Let's see square root of for l squared Less d squared minus for L. D. Co Cynthia minus four l d co sign. Okay, so then the question becomes after I saw for Q three Pecos and data over to sign of that. Okay, after I did all of that did I put it in the calculator correctly. Three p. Co sign 70 over to sign two times 70 minus 90. Plus The inverse sign of the sign of 70 times. De is too square root of four times 1.5 squared, plus two squared minus four times, 1.5 times two. Okay, Cool. Um, I don't see any mistakes there. Last thing I want to check because I'm not sure it makes sense. I'm seeing five less data minus 90 plus data, which is the same as Phi plus tooth eight of minus 90. What is five plus tooth? Eight of minus 90 and going to copy that Tasted here. I'm seeing that five plus tooth. Eight of minus 90 is 89 degrees. And that makes sense because I wanted to see if it was an acute angle. It is a cute just barely acute. So that makes sense. And so I'm going with everything. Makes sense

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